# Happy Birthday Kaprekar

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Today is 110th birthday of great Indian mathematician D. R. Kaprekar

Dattaraya Ramchandra Kaprekar (17 January 1905 – 1986) was an Indian recreational mathematician who described several classes of natural numbers. For his entire career (1930–1962) he was a schoolteacher at Nasik in Maharashtra. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties. Kaprekar was once laughed at by most contemporary Indian mathmaticians for his so-called ‘trivial’ play with numbers. It required G. H. Hardy to recognize Ramanujan while Kaprekar’s recognition came through Martin Gardner (he wrote about Kaprekar in his “Mathematical Games” column in March 1975 issue of “Scientific American”)

Here I would discuss a mathemagical trick re-disvovered by Kaprekar called “Gap Filling Process” (though claimed to be present in vedic mathematics)

Gap filling process

This process is magical one and will make you Mathemagician

Let, $(a)_n$ stand for $a$ repeated $n$ times (called Repunit $a$).

Then, we shall denote $(a)_n ^m$ for $m$-th power of $(a)_n$

$(9)_n ^m$ can be obtained by remembering the expansion for $(9)^m$ and inserting in the gaps between digits of expansion of $(9)^m$ with the numbers $(9)_{n-1}$ and $(0)_{n-1}$ alternately, beginning from left to right. No gap is counted after the unit digit.

Let’s see an example:
Find the value of $(99999)\times (99999)\times (99999) = (99999)^3 = (9)_5 ^3$.

Even my scientific calculator fails to calculate this exact value !

We know $9^3 = 729$

Then the gaps are: $----7----2----9$
Now fill the blanks alternately with $(9)_{5-1} = (9)_4$ and $(0)_{5-1} = (0)_4$.
We get: $9999\textbf{7}0000\textbf{2}9999\textbf{9}$
Hence, $(99999)\times (99999)\times (99999) = 999970000299999$

### 4 responses

1. This method is valid only for m = 1,2,3,4,5 in $9_n^m$. For higher powers we need to make alterations using the philosophy behind DEMLO NUMBERS (i.e. using digit sum in between)

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2. The above blog post is wrong.
Pardon me for typo.

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• As pointed out by Avinash Anand, this method is NOT universally valid. (Time for Discussion)
For example: the above method needs some alterations for $(9)_{16}^6$.
If you discover it please let me know.

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